We use the DMRG method to calculate several energy eigenvalues of the frustrated
S
= 1
/
2
square-lattice
J
1
-
J
2
Heisenberg model on 2
L
×
L
cylinders with
L
≤
10. We identify excited-level
crossings versus the coupling ratio
g
=
J
2
/J
1
and study their drifts with the system size
L
. The
lowest singlet-triplet and singlet-quintuplet crossings converge rapidly (with corrections
∝
L
−
2
)
to different
g
values, and we argue that these correspond to ground-state transitions between the
N ́eel antiferromagnet and a gapless spin liquid, at
g
c
1
≈
0
.
46, and between the spin liquid and a
valence-bond-solid at
g
c
2
≈
0
.
52. Previous studies of order parameters were not able to positively
discriminate between an extended spin liquid phase and a critical point. We expect level-crossing
analysis to be a generically powerful tool in DMRG studies of quantum phase transitions.